Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #87588⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #87588⋊C2 |
| Fitting | C22×#120 ⋊ D3 |
| derived | C22⋊(C22⋊A4) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊A4) | C22 |
| Frattini | C22 | C2×#1020 |
| Fitting | C22×#120 | D3 |
| socle | C24 | C2×S4 |
| LCS | G ▷ C22⋊(C22⋊A4) | |
| UCS | C1 | |
1080 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 16 |
Polycyclic presentation:
2^2,3,2^6|c^a=c^2,d^a=de,f^a=fg,h^a=hi,f^b=fh,g^b=gi,d^c=e,e^c=de,f^c=g,g^c=fg,h^c=i,i^c=hi30 conjugacy classes — view the full character table.
| maximal subgroup of | #72209642, #72215837, #72215860, … |
| maximal normal subgroup of | #72209642, #72215837, #72215860, … |
| maximal quotient of | #72209653, #72209655, #72209660, … |
| outer automorphism group of | C22×D4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090070);